Download Feynman-Kac formula for L´evy processes and semiclassical (Euclidean) momentum representation

Feynman-Kac formula for Lévy processes
and semiclassical (Euclidean) momentum
representation
Nicolas Privault∗
School of Physical and Mathematical Sciences
Nanyang Technological University
21 Nanyang Link, Singapore 637371
Xiangfeng Yang†
Department of mathematics
Linköping University
SE-581 83 Linköping, Sweden
Jean-Claude Zambrini‡
Grupo de Fı́sica Matemática - Universidade de Lisboa
Av. Prof. Gama Pinto 2
1649-003 Lisboa, Portugal
June 12, 2014
Abstract
We prove a version of the Feynman-Kac formula for Lévy processes and
integro-differential operators, with application to the momentum representation
of suitable quantum (Euclidean) systems whose Hamiltonians involve Lévytype potentials. Large deviation techniques are used to obtain the limiting
behavior of the systems as the Planck constant approaches zero. It turns out
that the limiting behavior coincides with fresh aspects of the semiclassical limit
of (Euclidean) quantum mechanics. Non-trivial examples of Lévy processes are
considered as illustrations and precise asymptotics are given for the terms in
both configuration and momentum representations.
Keywords and phrases: Lévy process, Feynman-Kac type formula, momentum representation, large deviations
AMS 2010 subject classifications: Primary 60J75, 60G51; secondary 60F10, 47D06.
∗
[email protected]
[email protected]
‡
[email protected]
†
1
1
Introduction
Consider the Lévy-type potential
Z
1 2 2
(e−ixk − 1 + ixk1{|k|≤1} )ν(dk),
x ∈ R,
(1.1)
V (x) = ibx + σ x −
2
R\{0}
R
where b, σ ∈ R and the Lévy measure ν satisfies R\{0} (1 ∧ k 2 )ν(dk) < ∞, and the
pseudo-differential operator V (i∇) given by
Z
σ 2 00
0
V (i∇)u(p) = −bu (p) − u (p) −
(u(p + k) − u(p) − ku0 (p)1{|k|≤1} )ν(dk),
2
R\{0}
(1.2)
cf. [12] and the references therein. In this paper we consider the partial differential
equation


 ∂u (t, p) = −U (p)u(t, p) − V (i∇)u(t, p), (t, p) ∈ (0, ∞) × R
∂t

u(0, p) = g(p),
p∈R
(1.3)
with prescribed initial condition g(p) and we derive the Feynman-Kac type formula
Z t
U (ξs ) ds ξ0 = p ,
(1.4)
u(t, p) = E g(ξt ) exp −
0
for the solution u(t, p) of (1.3), cf. Theorem 2.1 in Section 2.
Here, (ξt )t∈R+ is a real-valued Lévy process on (Ω, F, P), whose characteristic function
is given by the Lévy-Khintchine formula
E [exp (−ix · ξt )] = exp (−tV (x)) ,
with
where
σ2
V (x) = ibx + x2 −
2
R
R\{0}
Z
(1.5)
(e−ixk − 1)ν(dk),
R\{0}
(1 ∧ k 2 )ν(dk) < ∞ and the Lévy measure ν(dk) will be chosen symmetric
under time reversal k → −k so that (1.2) reduces to
Z
σ 2 00
0
(u(p + k) − u(p))ν(dk).
V (i∇)u(p) = −bu (p) − u (p) −
2
R\{0}
2
The function U (p) in (1.3) is sometimes called killing rate (see Section 3.5 in [1]). We
shall, however, avoid the term ‘killing’ which is inappropriate to our time reversible
framework, and refer simply to the ‘rate’.
Note that formulas such as (1.4) have been implicitly used in the literature, for example for the construction of subsolutions and the derivation of blow-up criteria for
semilinear PDEs of the form


 ∂u (t, p) = u1+β (t, p) − V (i∇)u(t, p), (t, p) ∈ (0, ∞) × R
∂t

u(0, p) = g(p),
p ∈ R,
where g is a nonnegative function, β > 0, see e.g. [4], [17], when V (i∇) = −(−∆)α/2
is a fractional Laplacian power, 0 < α ≤ 2, and [16] for the generator
Z
e−k
(u(p + k) − u(p))
V (i∇)u(p) = −
dk,
k
R\{0}
of the standard gamma process.
In this paper, based on Feynman and Hibbs’ suggestion that in the momentum representation of (Euclidean) quantum physics, the underlying stochastic processes should
belong to some class of time reversible jump processes (cf. [10]), we apply our result
to the momentum representation
Ĥu(p) = FHF−1 u(p)Ĥu(p) =
p2
u(p) + V (i~∇)u(p),
2
of the original (configuration representation) Hamiltonian
H=−
~2
4 + V,
2
where ~ is the Planck constant and F is the Fourier transform
Z
1
Fu(p) = √
e−ipx/~ u(x)dx,
2π~ R
for any rapidly decreasing function u(·) ∈ S(R), with inverse
Z
1
−1
F v(x) = √
eipx/~ v(p)dp,
v(·) ∈ S(R).
2π~ R
3
(1.6)
The corresponding Euclidean version of the associated Schrödinger equation in momentum representation (in the sense of [22] and [21]) reads
−~
∂ η̂ ∗
(t, p) = Ĥ η̂ ∗ (t, p),
∂t
(1.7)
which is one of the two adjoint equations whose positive solutions are needed to produce quantum-like probability measures in [22] and [21].
In addition we will consider a general rate function U (p) instead of p2 /2 in (1.6) and
deal with the equation

∗

~ ∂η (t, p) = −U (p)η ∗ (t, p) − V (i~∇)η ∗ (t, p), (t, p) ∈ (0, ∞) × R
∂t

η ∗ (0, p) = g(p),
p ∈ R,
(1.8)
instead of (1.7) with initial condition g(·) ∈ S(R). Finally we will study the limiting
behavior of (1.8) as ~ approaches 0.
Usually there are two different methods to derive a Feynman-Kac type formula:
• martingale methods combined with the Itô formula (see for example Section 5.7
of [14] and Section 3.19 of [23]), or
• semigroup methods (see Section 6.7.2 in [1], Section 3.19 in [23] and Section 1.5
of [11]).
The first method is generally applicable under the main assumption that the partial integro-differential equation (1.3) has a solution with suitable growth assumption
(see Proposition 4 in [18] and Theorem 4.1 in [6] for more details by using stochastic
integrals driven by Lévy processes). However, an unbounded rate U (p) function is
not compatible with the conditions of [18] and [6]. In this paper, we use the second
(semigroup) method to prove the existence (and even the uniqueness) of a solution
for (1.3) and to induce a Feynman-Kac type formula for the solution simultaneously.
It is also known that the semigroup method (with infinitesimal generator) only works
for (at least) bounded rates. In order to deal with the unbounded rate, in this paper
4
we use this method in a weaker form (explained at the beginning of Section 2) which
is explicitly presented in Theorem 2.1 and Theorem 2.3.
In addition we also study the limiting behaviors of the solutions given by FeynmanKac type formulas in terms of large deviations. In particular we show in detail that
these limiting behaviors have exactly the same patterns as the semiclassical limits of
(Euclidean) quantum mechanics in several special cases, and we derive precise asymptotics as well for the drift terms in both configuration and momentum representations.
The reader is referred to [28] for more about the underlying notion of Euclidean quantum mechanics.
This paper is organized as follows. In Section 2 we present the Feyman-Kac formula
(1.4) and its proof. The limiting behaviors of the solution given by Feynman-Kac
type formula are then studied in terms of large deviations in Section 3.
2
Feynman-Kac type formulas
Let A be the infinitesimal generator of the semigroup
Tt f (x) := E[f (ξt )|ξ0 = x],
of a time-homogeneous Markov process ξt where f is in a space B of bounded measurable functions with the uniform norm, and such that
Tt f (x) − f (x)
lim+ sup − Af (x) = 0,
t→0
t
x
f ∈ B.
For every f in the domain of A, the function
u(t, x) := Tt f (x) = Ex [f (ξt )]
is the unique (in the sense of boundedness) solution of
∂u
(t, x) = Au(t, x) with u(0, x) = f (x).
∂t
5
The main ideas for the derivation of Feynman-Kac type formulas by semigroup methods for differential equations of the form
∂v
(t, x) = Av(t, x) − c(x)v(t, x), v(0, x) = f (x)
∂t
(2.1)
with rate c(x), can be formulated as follows, see Section 3.19 of [23] and Section 1.5
of [11] for details.
The idea of the semigroup method is to show that
e (x) := Af (x) − c(x)f (x)
Af
is the infinitesimal generator of the new semigroup
Z t
e
Tt f (x) := Ex f (ξt ) exp −
c(ξs )ds ,
0
i.e. v(t, x) := Tet f (x) is a solution of (2.1). This only holds under (strong) assumptions,
including the boundedness of c(x), cf. [23]. Note that in this case the derivative
∂v
(t, x) = Av(t, x) − c(x)v(t, x)
∂t
is uniform in x, in the sense that
v(t + h, x) − v(t, x)
lim sup − (Av(t, x) − c(x)v(t, x)) = 0,
h→0 x
h
(2.2)
for every fixed t > 0. However, pointwise convergence is sufficient in (2.2) for the
analysis of a differential equation, and we choose to weaken the boundedness condition on c(x) when deriving our Feynman-Kac formulas.
In the sequel, (ξt )t≥0 will be a Lévy process defined on (Ω, F, P) whose characteristic
exponent is given by (1.5), and which is Markov with respect to the increasing filtration Pt = σ(ξs , 0 ≤ s ≤ t). Furthermore, (ξt )t≥0 has a modification (still denoted
by (ξt )t≥0 ) which is right continuous with left limits. We say that the process ξ has
bounded jumps if there is C > 0 such that
sup |ξt − ξt− | < C
t≥0
6
a.s.
It follows from Corollary 2.4.9 of [1] that ξ has bounded jumps if and only if ν has a
bounded support.
We denote by
S(R, C) := f (·) : R → C with sup xm f (n) (x) < ∞,
x
∀ m, n ∈ N ,
the Schwartz space of all infinitely differentiable rapidly decreasing functions with
bounded n-th derivatives f (n) (x) on the real line R to the complex plane C, with
S(R) = S(R, R).
Theorem 2.1. Assume that the rate function U (p) is smooth and such that
U (p) ≥ c|p|a ,
|p| > C,
(2.3)
for some c > 0, a ≥ 1 and C > 0, and that its derivatives satisfy
(n) U (p) ≤ c(n) 1 + |p|k(n) ,
|p| > C,
(2.4)
for some c(n), k(n) > 0. Then (1.4) is a solution to (1.3) provided ν has moments of
all orders and g ∈ S(R).
Proof. Throughout this proof we take U (p) = p2 /2 without loss of generality. For any
bounded measurable f (x), let Tt f (x) = E[f (ξt + x)] denote the semigroup associated
with the Lévy process ξt started at ξ0 = 0. For any function g ∈ S(R), the infinitesimal
generator A of ξt is
σ2
Ag(x) = bg (x) + g 00 (x) +
2
0
Z
(g(x + k) − g(x)) ν(dk),
R\{0}
see Theorem 3.3.3 in [1]. We also define
Z
1 t
2
e
Tt g(x) = E g(ξt ) exp −
|ξs | ds ξ0 = x
2 0
e
and let Ag(x)
:= Ag(x) − x2 g(x)/2.
7
Step 1: We show for each fixed x ∈ R and g ∈ S(R) that
e
lim+ t−1 Tet g(x) − g(x) = Ag(x).
(2.5)
t→0
By the relation
Z
Tet g(x) − g(x) =
t
e
Tes Ag(x)ds
(2.6)
0
which follows from martingale arguments, cf. e.g. Section 3.19 of [23], we get
Z t
−1 e
−1
e
lim t
Tes Ag(x)
Tt g(x) − g(x) = lim t
t→0+
t→0+
0
Z
1 s
1 2
−1
2
|ξu | du ξ0 = x ds
E Ag(ξs ) − |ξs | g(ξs ) exp −
= lim+ t
t→0
2
2 0
0
Z 1 Z
1
1 tθ
2
2
= lim+
E Ag(ξtθ ) − |ξtθ | g(ξtθ ) exp −
|ξu | du ξ0 = x dθ.
t→0
2
2 0
0
Z
t
The first term is easily to be computed as
Z 1 Z
1 tθ
2
lim
E Ag(ξtθ ) exp −
|ξu | du ξ0 = x dθ = Ag(x)
t→0+ 0
2 0
since Ag(x) is bounded and continuous in x. The second limit
Z 1 Z
1 tθ
2
2
lim
E ξtθ · g(ξtθ ) exp −
|ξu | du ξ0 = x dθ = x2 g(x)
t→0+ 0
2 0
(2.7)
since x2 g(x) is bounded and continuous in x which is from the fact that g ∈ S(R).
However, here we present another proof of (2.7) which works for any (only) bounded
and continuous g. The reason why we give this proof is that, in Theorem 2.3 below,
the initial g is only assumed to be bounded and continuous, and such a proof is needed
there. Under the existence of moments of all orders for ν we have the moment bounds
sup E [(ξt )m | ξ0 = x] < ∞,
m ≥ 1,
t∈[0,1]
cf. e.g. [2] and Relations (1.1)-(1.2) in [19]. We write
Z 1 Z
1 tθ
2
2
lim
E |ξtθ | g(ξtθ ) exp −
|ξu | du ξ0 = x dθ
t→0+ 0
2 0
Z 1 h
i
2
= lim+
E |ξtθ | g(ξtθ ) ξ0 = x dθ
t→0
0
8
(2.8)
1
Z
+ lim+
t→0
0
Z
1 tθ
2
2
|ξu | du − 1 ξ0 = x dθ
E |ξtθ | g(ξtθ ) exp −
2 0
=: I + II,
with I = x2 g(x) because of (2.8). The second component vanishes by noting that
#
"
2 Z
1 tθ
E exp −
|ξu |2 du − 1
ξ0 = x dθ = o(t).
2 0
Step 2: We show that Tet g(·) ∈ S(R) for all g ∈ S(R) and t > 0. From the independence and stationarity of increments of ξt we have
Z
1 t
2
e
|ξs | ds ξ0 = x
Tt g(x) = E g(ξt ) exp −
2 0
Z
1 t
2
= E g(ξt + x) exp −
|ξs + x| ds ,
2 0
hence by the bound E[|ξt |k ] < ∞, k ≥ 1, the n-th partial derivatives ∂ n Tet g(x)/∂xn
can be bounded as
∂ n Tet g(x)
∂xn
!2
≤ c(n)(1 + x
2k(n)
Z
1 t
2
)E exp −
|ξs + x| ds
2 0
(2.9)
for some positive constant c(n) depending on n and an integer k(n) depending on n.
For any positive integer m, we analyze the part with expectation in (2.9) as follows
"
2 !#
Z
Z t
1 t
t
1
E exp −
|ξs + x|2 ds
≤ E exp −
x+
ξs ds
2 0
2
t 0
"
2 !#
Z t
t
1
x+
ξs ds
(2.10)
= E 1{ω:|R t ξs ds|≤t|x|1/2 } exp −
0
2
t 0
"
2 !#
Z t
t
1
+ E 1{ω:|R t ξs ds|>t|x|1/2 } exp −
x+
ξs ds
=: `1 + `2 .
0
2
t 0
For `2 , it follows from Tchebychev type estimates that
`2 ≤ P
"Z
Z t
2m #
t
1
1
ω : ξs ds > |x|1/2
≤ m 2m E
ξs ds
.
t 0
|x| t
0
9
Thus supx |x|m `2 < ∞ holds since E[(ξt )k ] < ∞. It is easy to see that for large enough
|x| we have `1 ≤ e−t|x|/2 , which yields
∂ n Te g(x) m
t
sup x
< ∞.
n
∂x
x Step 3: In this step, we replace g(x) in (2.5) by Tet g(x) to get
∂ + Tet g(x)
eTet g(x),
=A
∂t
Te0 g(x) = g(x).
In this step, we show that the right-hand derivative can be replaced by the two-sided
derivative ∂ Tet g(x)/∂t. To this end, we just need to show that the right-hand derivative
eTet g(x) is continuous in t. Let us recall
A
eTet g(x) = ATet g(x) − 1 x2 Tet g(x).
A
2
The continuity in t is then easily from the definitions of A and Tet with the help of
(2.6) repeatedly.
Note that in Theorem 2.1 the condition (2.3) (i.e. U (p) ≥ c|p|a for some a ≥ 1)
is necessary in general since in the first inequality of (2.10) we need the convexity
of c|p|a in p. However, condition (2.3) can be weakened to any a > 0 by assuming
in addition that the process (ξt )t≥0 is a subordinator, i.e. a one-dimensional a.s.
non-decreasing Lévy process, cf. Section 1.3.2 in [1], which will remain a.s. positive
provided ξ0 ≥ a∗ > 0. For instance, Theorem 2.2 applies in case U (p) = p1/2 .
Theorem 2.2. Assume that (ξt )t≥0 is a subordinator and that in addition to (2.4) the
smooth rate function U (p) satisfies
U (p) ≥ c|p|a ,
|p| > C,
for some c > 0, a > 0 and C > 0. Then (1.4) is a solution to (1.3) provided ν has
moments of all orders and g ∈ S(R).
10
Proof. We adjust (2.10) to make the arguments go through for the new (possibly not
convex) function cpa . Namely we write
Z t
Z t
a
E exp −
U (ξs + x) ds
≤ E exp −c
(ξs + x) ds
0
0
Z t
a
= E 1{ω: sup0≤s≤t ξs ≤x1/2 } exp −c
(ξs + x) ds
0
Z t
a
+ E 1{ω: sup0≤s≤t ξs >x1/2 } exp −c
(ξs + x) ds
:= `1 + `2 ,
(2.11)
x > 0,
0
with `1 ≤ exp −ctxa/2 for large x as above. The second term is estimated as
`2 ≤ P ω : sup0≤s≤t ξs > x1/2 = P ω : ξt > x1/2 , then Tchebychev type estimates
complete the argument.
The condition g ∈ S(R) in Theorem 2.1 is somewhat restrictive and can also be
relaxed into assuming only continuity and boundedness of g if we restrict ourselves to
a special family (ζt )t≥0 of Lévy processes which are real-valued pure jump processes
with infinitesimal generator
Af (p) =
Z
(f (p + k) − f (p))µ(dk)
R
for bounded f (·) : R → R, where µ(R) < ∞ and µ(dk) is symmetric under k → −k,
cf. e.g. Section 4.2 of [9]. In this setting we consider the partial integro-differential
equation
Z

∂u∗


(t, p) = −U (p)u∗ (t, p) + (u∗ (t, p + k) − u∗ (t, p))µ(dk), (t, p) ∈ (0, ∞) × R
∂t
R

u (0, p) = g(p), p ∈ R,
∗
(2.12)
where U (p) is continuous and satisfies c1 ≤ U (p) ≤ c2 (1+|p|M ) for some c1 ∈ R, c2 > 0
and M > 0. This allows us in particular to work in the momentum representation of
(Euclidean) quantum mechanics with a general potential U (p) instead of only p2 /2 as
in [21].
11
Theorem 2.3. Suppose that µ(dk) is symmetric under k → −k, µ(R) < ∞, and
U (p) is continuous and satisfies
c1 ≤ U (p) ≤ c2 (1 + |p|M )
for some c1 ∈ R, c2 > 0 and M > 0. If the initial condition g of (2.12) is continuous
R
and bounded, and R |k|2M µ(dk) < ∞, then
Z t
u∗ (t, p) = E g(ζt ) exp −
U (ζs ) ds ζ0 = p
(2.13)
0
is a solution to (2.12).
Proof. The proof is quite similar to that of Theorem 2.1 by replacing S(R) with the
space of bounded measurable functions on R. Since µ(R) is finite the finiteness of the
R
moment supt∈[0,1] E[(ζt )2M ] follows directly from the assumption R |k|2M µ(dk) < ∞
by e.g. [2] or Relations (1.1)-(1.2) in [19]. In Steps 2 and 3 we need the boundedness
and the continuity in x of the new semigroup
Z t
e
U (ζs ) ds ζ0 = p
Tt g(p) = E g(ζt ) exp −
0
which follow from the boundedness and the continuity of g(x).
There are a number of ways to prove that (2.13) is the unique (bounded) solution
to (2.12) under additional appropriate assumptions. For instance, if the rate U is
bounded, then the Gronwall’s lemma implies the uniqueness.
3
Limiting behaviors
This section is concerned with the limiting behavior of solutions and drift terms as ~
tends to 0. We start in Section 3.1 with an illustration of how large deviations (of integral forms) can be applied in the case of pure jump processes. Detailed formulations
are then presented together with some close connections with the semiclassical limits
discussed in [15]. Finally we provide precise asymptotics for the drift terms in both
configuration and momentum representations of Euclidean quantum physics. See also
[20] for recent large deviations results for continuous Bernstein processes, and [27] in
the case of jump processes.
12
3.1
An illustration with pure jump processes
To illustrate the analysis of limiting behaviors of solutions defined through (1.4) and
(2.13) we consider the limit as ~ tends to 0 of the solution
Z t
~
~
−1
~
~
U (ζs ) ds ζ0 = p .
u (t, p) = E exp −~
0
to (2.12) when g ≡ 1 and (ζt~ )t≥0 is the pure jump processes with infinitesimal generator
Af (p) = ~
−1
Z
(f (p + ~k) − f (p))µ(dk).
R
Besides the time symmetry assumption and µ(R) < ∞, we further impose the bounded
support condition
µ([−N, N ]c ) = 0, µ([−N, −N + ]) > 0, µ([N − , N ]) > 0, ∀ > 0,
on the measure µ, for some N > 0. The trajectories of (ζs~ )0≤s≤t belong to the space
D([0, t]) of all right continuous functions with left limits (equipped with the uniform
norm). It has been shown in [25] (see Theorems 3.2.1, 3.2.2 and 4.1.1 therein) that
the family (ζs~ )0≤s≤t satisfies a large deviation principle over D([0, t]) with the action
Rt
functional S(φ) := 0 L0 (φ0 (s))ds for absolutely continuous function φ(t) where
Z
L0 (u) = sup(xu − H0 (x)), H0 (x) = (exk − 1)µ(dk).
x∈R
R
More precisely, for any measurable set Γ ⊆ D([0, t]) we have
− info S(φ) ≤ lim inf ~ ln P~ (ζ ~ ∈ Γ) ≤ lim sup ~ ln P~ (ζ ~ ∈ Γ) ≤ − inf S(φ)
φ∈Γ
~→0
~→0
(3.1)
φ∈Γ̄
where Γo (resp. Γ̄) is the interior (resp. closure) of Γ.
It is proved from (3.1) through Varadhan’s integral lemma (see Section 4.3 [8]) that
Z t
~
~
−1
~
lim ~ ln E exp −~
U (ζs ) ds ζ0 = p
~→0
0
Z t
=
sup
−
U (φ(s) + p) ds − S(φ)
φ∈D0 ([0,t])
0
13
=
Z t
0
L(φ (s), φ(s))ds ,
−
sup
φ∈D0 ([0,t])
0
where L(φ0 (s), φ(s)) = L0 (φ0 (s))+U (φ(s)+p) and D0 ([0, t]) is the subspace of D([0, t])
with initial position 0. For general g satisfying suitable boundedness and smoothness
conditions, we may also deduce the asymptotic expansion
Z
!
t
u~∗ (t, p) = exp h−1
sup
L(φ0 (s), φ(s)) ds
−
φ∈D0 ([0,t])
0
!
X
ki ~i/2 + o(~n/2 )
0≤i≤n
as ~ → 0, by means of precise asymptotics for large deviations which have been
established in [26] and Chapter 5 of [25].
3.2
Harmonic oscillator Hamiltonian
In this section we turn to (1.4) and take
U (p) = p2 /2,
b = ν = 0,
and σ 2 = 1
in the characteristic exponent (1.5), i.e. (ξt~ )t∈R+ =
√
~(Wt )t∈R+ , where (Wt )t∈R+ is a
one-dimensional Wiener process on the real line.
In order to show the connection between large deviations and semiclassical limit of
(Euclidean) quantum mechanics we consider ū~ (t, p) := u~∗ (1 − t, p) for t ∈ [0, 1]. It
follows from (1.3) with initial g(p) = exp (−p2 /~) that the positive function ū~ (t, x)
satisfies the final value problem

~
2

~ ∂ ū (t, p) = − ~ 4ū~ (t, p) + 1 p2 ū~ (t, p), (t, p) ∈ [0, 1) × R
∂t
2
2

ū~ (1, p) = exp (−p2 /~) ,
p ∈ R.
(3.2)
From Schilder’s large deviation formula (see [24]), ū~ (t, x) admits the asymptotic
expansion
Z 1 1
1 0
1
2
2
2
ū (t, p) = exp −
|q̄ (τ )| + |q̄(τ ) + p| dτ + |q̄(1) + p| + o(1)
~
2
2
t
~
(3.3)
14
as ~ → 0, where q̄(τ ) := φ̄(τ − t) and φ̄ is a minimizer of
Z 1−t 1 0
1
2
2
2
inf
|φ (s)| + |φ(s) + p| ds + |φ(1 − t) + p|
φ∈A0 ([0,1−t])
2
2
0
with A0 ([0, 1 − t]) being the space of all absolutely continuous functions on [0, 1 − t]
having initial value 0. In terms of q̄ itself, the variational problem can therefore be
rewritten as on [t, 1] :
Z
1
inf
q∈A0 ([t,1])
t
1 0
1
2
2
2
|q (s)| + |q(s) + p| ds + |q(1) + p| .
2
2
If φ̄ is smooth, then it satisfies an ordinary differential equation
φ̄00 (s) − (φ̄(s) + p) = 0, s ∈ (0, 1 − t),
φ̄(0) = 0 and φ̄0 (1 − t) + 2(φ̄(1 − t) + p) = 0.
It is again equivalent but more appropriate for us to rewrite this in terms of q̄ as
q̄ 00 (s) − (q̄(s) + p) = 0, s ∈ (t, 1),
q̄(t) = 0 and q̄ 0 (1) + 2(q̄(1) + p) = 0.
We note that (3.3) has a quite similar expression as the semiclassical limit (5.38) in
[15]. Actually, we can match exactly these two expressions by investigating the exact
√
order o(1) in (3.3). By applying precise large deviations of [24] to ~W, we get the
expansion
ū~ (t, p) = (1 + o(1))K̄(t)
(3.4)
Z 1 1
1 0
1
× exp −
|q̄ (τ )|2 + |q̄(τ ) + p|2 dτ + |(q̄(1) + p|2
~
2
2
t
h
R
i
1−t
2
as ~ → 0, where K̄(t) = E exp − 21 0 Ws2 ds − W1−t
. In order to compare (3.4)
with (6.8)-(6.9) in [15], we note that K̄(t) = (2πF (t))−1/2 where
F (t) =
1
(cosh(1 − t) + 2 sinh(1 − t))
2π
solving the following ordinary differential equation on [t, 1] (cf. (1.9.3) page 168 in
[5])
F 00 (τ ) = F (τ ),
τ ∈ [t, 1],
F (1) = 1/(2π),
15
F 0 (1) = −1/π.
In particular, these final boundary conditions satisfy the condition (6.9) of [15]:
|F 0 (1)| + |F (1)| > 0. This means that the associated (forward) semiclassical Bern√
stein diffusion Z of [15] is absolutely continuous with respect to our ξ ~ = ~W and
with linear drift B(z, τ ) = F 0 (τ )F −1 (τ )z (the constant δ, there, is zero). The semiclassical analysis done here for (3.2) is valid for Gaussian final boundary condition
ū~ (1, p).
Equation (3.2) is the forward description (involving the usual increasing filtration
representing the past information about the system) of the semiclassical expansion,
and we can similarly consider the following backward description

~
2

~ ∂u∗ (t, p) = ~ 4u~∗ (t, p) − 1 p2 u~∗ (t, p), (t, p) ∈ (0, 1] × R
∂t
2
2

u~ (0, p) = exp −p2 /~ ,
p∈R
(3.5)
associated with a decreasing filtration. For simplicity, the same Gaussian boundary
condition has been chosen. In this case, the solution u~∗ (t, p) admits the expansion
u~∗ (t, p) = K∗ (t) · (1 + o(1))
Z t 1
1 0
1
2
2
2
|q (τ )| + |q∗ (τ ) + p| dτ + |q∗ (t) + p|
× exp −
~
2 ∗
2
0
(3.6)
where q∗ is the minimizer of
Z t 1
1 0
2
2
2
inf
|q (s)| + |q(s) + p| ds + |q(t) + p| ,
q∈A0 ([0,t])
2
2
0
h
R
i
t
and the coefficient K∗ (t) = E exp − 21 0 Ws2 ds − Wt2 . Then we have K∗ (t) =
(2πF∗ (t))−1/2 , where F∗ solves the same ODE as F before but with initial boundary
conditions on [0, t] : F∗ (0) = 1/(2π) and F∗0 (0) = 1/π. Then F∗ (τ ) =
1
(cosh τ
2π
+
2 sinh τ ). The same semiclassical diffusion Z as before, now considered with respect
to a decreasing, or backward, filtration has a drift B∗ (z, τ ) = F∗0 (τ )F∗−1 (τ )z. Again, the
whole manifold of semiclassical diffusions follow from the analysis of general Gaussian
initial condition (cf. [15]). The special Gaussian Z defined via (3.4) and (3.6) is of
zero mean and covariance c(τ ) proportional to F∗ (τ )F (τ ), 0 ≤ τ ≤ 1.
16
3.3
Pure jump Hamiltonian
The operator H = −~2 4/2 + p2 /2 of equation (3.2) is called Harmonic oscillator
Hamiltonian expressed here, in this special case where configuration and momentum
play a completely symmetric role, in terms of the momentum variable. We can generalize (3.2) to include more general Hamiltonians and derive precise large deviations
as (3.4) having similar connections with the semiclassical limits of (Euclidean) quantum physics, see [15], [25] and [26]. In particular, we illustrate below an example
with pure jump processes as mentioned in Section 3.1. Consider the following partial
integro-differential equation with final condition
Z

∂v̄ ~

~
(t, p) = − (v̄ ~ (t, p + ~k) − v̄ ~ (t, p))µ(dk) + U (p)v̄ ~ (t, p), (t, p) ∈ [0, 1) × R
∂t
R

v̄ ~ (1, p) = exp (−1/~) , p ∈ R.
(3.7)
If we consider a special potential V (x) = 1 − cos(αx) for some α > 0, this case
corresponds to µ(dk) = 21 (δα (dk) + δ−α (dk)). For a particular U (p) = p2 − p, it has
been proved in [26] that v̄ ~ has the asymptotics
Z 1
1
~
0
v̄ (t, p) = P̄ (t) · (1 + o(1)) · exp −
L(z̄ (s), z̄(s))ds + 1
~
t
(3.8)
as ~ → 0, where L is defined in Section 3.1 with
Z 1
q
q
2
2
0
0
0
0
S(z) =
(z (s)/α) ln z (s)/α + (z (s)/α) + 1 + 1 − (z (s)/α) + 1 ds
t
(3.9)
for absolutely continuous z(s), the function z̄(τ ) is the unique minimizer of
Z 1
L(z 0 (s), z(s))ds
t
h
R
i
1
over A0 ([t, 1]), and P̄ (t) = E exp − t ς¯s2 ds with
r
ς¯s = α
e%̄(s−t) + e−%̄(s−t)
Ws−t
2
17
and
p
0
2
%̄(τ ) = ln z̄ (τ )/α + (z̄ (τ )/α) + 1 .
0
We further remark that the unique minimizer z̄ satisfies the following ordinary differential equation
z̄ 00 (s) − (2(z̄(s) + p) − 1) ·
p
|z̄ 0 (s)|2 + α2 = 0,
s ∈ (t, 1),
z̄(t) = 0 and z̄ 0 (1) = 0.
Again (3.8) is the forward description of the semiclassical expansion. Let us now
consider the backward description by using
Z

∂v∗~

~
(t, p) = (v∗~ (t, p + ~k) − v∗~ (t, p))µ(dk) − U (p)v∗~ (t, p), (t, p) ∈ (0, 1] × R
∂t
R

v ~ (0, p) = exp (−1/~) ,
p ∈ R.
∗
(3.10)
In this case, we have
Z t
1
0
= P∗ (t) · (1 + o(1)) · exp −
L(z∗ (s), z∗ (s))ds + 1
(3.11)
~
0
Rt
as ~ → 0, where z∗ is the unique minimizer of 0 L(z 0 (s), z(s))ds over A0 ([0, t]) (of
v∗~ (t, p)
course the action functional S defined by (3.9) now has the integral interval [0, t]),
and
Z t
2
P∗ (t) = E exp −
ςs ds
0
p
with ςs = α (e%∗ (s) + e−%∗ (s) )/2Ws and
p
%∗ (τ ) = ln z∗0 (τ )/α + |z∗0 (τ )/α|2 + 1 .
3.4
Precise asymptotics for drift terms
In [7], Bernstein processes were studied as stochastic deformation related to Feynman
path integral. These processes were first introduced in [3], and now are also called
local Markov processes, two-sided Markov processes or reciprocal process in literature,
see [13]. In configuration representations, basically they are diffusion processes over a
finite time interval with prescribed initial and final distributions. In the momentum
18
representation, the drift parts (terms) can be expressed as ~∇u~∗ (t, p)/u~∗ (t, p) with
u~∗ (t, p) defined in (1.3). In the free case (that is, the potential is identically zero), the
drift has an explicit form which is independent of ~ (cf. Section 5.1 in [7]). Our goal
in this section is to derive precise asymptotics as ~ → 0 for the drift terms in both
configuration and momentum representations. In order to have explicit comparisons
with the results in [7], here we adopt some new notations which are slightly different
from those in previous sections. For instance, the solution u~∗ (t, p) is now denoted as
η∗~ (t, q) or η̃∗~ (t, p), and the potential is still denoted by V (with different expressions)
which is fully explained each time when it is used.
3.4.1
Configuration representation Hamiltonian
We consider a Hamiltonian H = −~2 4/2+V in configuration representation, where V
is a potential in Kato’s class. The backward description of the semiclassical expansion
is


∂η ~
~2

~ ∗ (t, q) = 4η∗~ (t, q) − V (q)η∗~ (t, q), (t, q) ∈ (0, ∞) × R
∂t
2
1
2

~

e−q /(2~) , q ∈ R.
η∗ (0, q) = √
2π~
(3.12)
We aim at finding precise asymptotics for ~∇η∗~ (t, q)/η∗~ (t, q) as ~ → 0. To this end, we
introduce the space C0 ([0, t]) which consists of continuous functions with zero initial
value, C01 ([0, t]) whose elements are in C0 ([0, t]) and continuously differentiable, and
three functionals
Z
t
|φ(t) + q|2
V (φ(s) + q)ds +
2
F (φ) = −
0
Z t
G(φ) =
V 0 (φ(s) + q)ds + (φ(t) + q),
0
Z
1 t 0 2
S(φ) =
φ (s) ds,
2 0
,
for absolutely continuous φ ∈ C0 ([0, t]), and S(φ) = ∞ otherwise. Now we state the
main result of this section.
Proposition 3.1. Assume that
19
(A.1) the potential V (·) : R → R is bounded below and has up to fourth order
continuous derivatives with V 00 (q) ≥ 0, and
(A.2) the supremum of F − S over C0 ([0, t]) is reached uniquely at φ∗ ∈ C01 ([0, t]).
Then we have the equivalence
∇η∗~ (t, q)
K̄1∗
~ ~
∼ − G(φ∗ ) + ∗ ~ + o(~) ,
η∗ (t, q)
K0
as ~ → 0, where
1 00
∗
F (φ∗ )(W, W )
K0 = E exp
2
K̄0∗ = G(φ∗ ) · K0∗
1 00
1 00
1
3
∗
K̄1 = E exp
F (φ∗ )(W, W )
G (φ∗ )(W, W ) + G0 (φ∗ )(W )F (3) (φ∗ )(W ⊗ )
2
2
6
2 1
1
(4)
⊗4
(3)
⊗3
+ G(φ∗ )F (φ∗ )(W ) + G(φ∗ ) F (φ∗ )(W )
.
24
72
Remark 3.2. On the interval [0, t], the unique maximizer φ∗ satisfies the ordinary
differential equation
φ00∗ (τ ) − V 0 (φ∗ (τ ) + q) = 0, with φ∗ (0) = 0, φ0∗ (t) = −(φ∗ (t) + q),
with in particular G(φ∗ ) = −φ0∗ (0). In order to express the dependence of G(φ∗ ) on q
and the time t we can use the transform Φ∗ (τ ) = φ∗ (t − τ ) + τ · q and in this case
Φ00∗ (τ ) − V 0 (Φ∗ (τ ) − τ · q + q) = 0, with Φ∗ (t) = qt, Φ0∗ (0) = Φ∗ (0) + q
from which G(φ∗ ) = −φ0∗ (0) = Φ0∗ (t) − q.
Remark 3.3. In the formulation of Proposition 3.1, all functional derivatives are in
the sense of Fréchet derivative. For instance,
Z t
0
0
F (φ∗ )(W ) = −
V (φ∗ (s) + q)Ws ds + (φ∗ (t) + q)Wt
0
00
and F (φ∗ )(W, W ) = −
R
t
0
00
V (φ∗ (s) +
q)Ws2 ds
20
+
Wt2
.
Proof of Proposition 3.1. The solution to (3.12) can be written by a Feynman-Kac
formula as
η∗~ (t, q)
1
1 √
=√
E exp
F ( ~W ) ,
~
2π~
(3.13)
where the functional F is defined in Proposition 3.1. Then the derivative of η∗~ (t, q)
with respect to q is written as
∇η∗~ (t, q)
√
1
1
1 √
=− ·√
E G( ~W ) · exp
F ( ~W )
~
~
2π~
(3.14)
with the functional G as above. From precise asymptotics for large deviations of
integral forms (cf. [25] and [26]), expansions for (3.13) and (3.14) can be proved as
follows, respectively,
η∗~ (t, q)
1
1 √
=√
E exp
F ( ~W )
~
2π~
1
1
exp
(F (φ∗ ) − S(φ∗ )) · (K0∗ + o(1))
=√
~
2π~
(3.15)
and
∇η∗~ (t, q)
√
1
1
1 √
=− ·√
E G( ~W ) · exp
F ( ~W )
~
~
2π~
1
1
1
exp
(F (φ∗ ) − S(φ∗ )) · K̄0∗ + K̄1∗ ~ + o(~) ,
=− ·√
~
~
2π~
as ~ → 0. Combining (3.15) and (3.16), it follows that
− exp ((F (φ∗ ) − S(φ∗ )) /~) · K̄0∗ + K̄1∗ ~ + o(~)
∇η∗~ (t, q)
∼
~ ~
η∗ (t, q)
exp ~1 (F (φ∗ ) − S(φ∗ )) · (K0∗ + o(1))
K̄1∗
∼ − G(φ∗ ) + ∗ ~ + o(~) ,
K0
as ~ → 0.
3.4.2
Momentum representation Hamiltonian
Consider the complex-valued potential
Z
σ2q2
V (q) = ibq +
+
e−iqk − 1 + iqk1{|k|≤1} ν(dk)
2
R\{0}
21
(3.16)
which yields the pseudo-differential operator
2σ
0
V (i~∇)u(p) = −~bu (p) − ~
2
2
Z
00
u (p) −
(u(p + ~k) − u(p)) ν(dk)
R\{0}
where the Lévy measure ν(dk) is assumed to be symmetric under the time reversal
k → −k.
The resulting Hamiltonian Ĥ = p2 /2 + V (i~∇) is the momentum representation of
H = −~2 4/2 + V. Then the backward description is the following partial integrodifferential equation,

~

~ ∂ η̃∗ (t, p) = − 1 p2 η̃∗~ (t, p) − V (i~∇)η̃∗~ (t, p), (t, p) ∈ (0, ∞) × R
∂t
2

η̃ ~ (0, p) = √ 1 e−p2 /(2~) , p ∈ R.
∗
(3.17)
2π~
Because of the jumps, in this section we consider the space D0 ([0, t]). In order to obtain
the precise asymptotics for ~∇η̃∗~ (t, q)/η̃∗~ (t, q) as ~ → 0, we define three functionals
over D0 ([0, t]) :
1
F̃ (φ) = −
2
Z
t
1
|φ(s) + p|2 ds − |φ(t) + p|2 ,
2
0
Z t
G̃(φ) = φ(t) + p +
(φ(s) + p)ds,
0
Z t
L0 (φ0 (s))ds,
S̃(φ) =
0
for absolutely continuous φ, where L0 (u) = supx∈R (xu − H0 (x)) and
Z
σ 2 x2
H0 (x) = bx +
+
(exk − 1)ν(dk),
2
R\{0}
and S(φ) = ∞ otherwise.
The operator V (i~∇) is associated with a family of Lévy processes which satisfies a
large deviation principle under suitable assumptions on H0 , cf. Conditions (A)-(E) in
Sections 3.1 and 3.2 of [25], which are assumed to hold in the following proposition.
22
Proposition 3.4. If the supremum of F̃ − S̃ over D0 ([0, t]) is reached uniquely at
φ̃∗ ∈ C01 ([0, t]), then following precise asymptotics holds:
~
∇η̃∗~ (t, p)
∼ −G̃(φ̃∗ ) + o(~),
η̃∗~ (t, p)
as ~ → 0.
Remark 3.5. The unique maximizer φ̃∗ satisfies an ordinary differential equation on
[0, t] :
d 0
L0 (u)|u=φ̃0∗ (τ ) = φ̃∗ (τ ) + p, with φ̃∗ (0) = 0, L00 (u)|u=φ̃0∗ (t) = −(φ̃∗ (t) + p),
dτ
from which it follows G̃(φ̃∗ ) = −L00 (u)|u=φ̃0∗ (0) .
Proof of Proposition 3.4. The solution to (3.17) is written as a Feynman-Kac formula
1
1
~
~
~
η̃∗ (t, p) = √
E exp
F̃ (ξ ) ,
(3.18)
~
2π~
where (ξt~ )t∈R+ defined on (Ω, F, P~ ) is a real-valued Lévy process for each fixed ~
whose characteristic function is given by the Lévy-Khintchine formula
i
t
~
~
E exp − · x · ξt
= exp − V (x)
~
~
R
2
with V (x) = ibx + σ2 x2 − R\{0} e−ixk − 1 ν(dk). The derivative of η̃∗~ (t, p) with
respect to p is
∇η̃∗~ (t, p)
1
1
1
~
~
~
=− ·√
E G̃(ξ ) · exp
F̃ (ξ ) .
~
~
2π~
It is again from [25] and [26] that expansions for (3.18) and (3.19) are
1
1
~
~
~
η̃∗ (t, p) = √
E exp
F̃ (ξ )
~
2π~
1
1
=√
exp
F̃ (φ̃∗ ) − S̃(φ̃∗ )
· (K0 + o(1))
~
2π~
(3.19)
(3.20)
and
∇η̃∗~ (t, p)
1
1
1
~
~
~
=− ·√
E G̃(ξ ) · exp
F̃ (ξ )
~
~
2π~
1
1
1
=− ·√
exp
F̃ (φ̃∗ ) − S̃(φ̃∗ )
· K̄0 + K̄1 ~ + o(~) ,
~
~
2π~
23
(3.21)
h
i
where K0 = E exp 21 F̃ 00 (φ̃∗ )(ζ, ζ) , K̄0 = G̃(φ̃∗ ) · K0 ,
1 00
1 00
1
3
K̄1 = E exp
F̃ (φ̃∗ )(ζ, ζ)
G̃ (φ̃∗ )(ζ, ζ) + G̃0 (φ̃∗ )(ζ)F̃ (3) (φ̃∗ )(ζ ⊗ )
2
2
6
2 1
1
(4)
⊗4
(3)
⊗3
+ G̃(φ̃∗ )F̃ (φ̃∗ )(ζ ) + G̃(φ̃∗ ) F̃ (φ̃∗ )(ζ )
=0
24
72
R
for ζ being a diffusion process with diffusion coefficient σ 2 + R\{0} ez∗ (t)k k 2 ν(dk) and
z∗ (t) = L00 (u)|u=φ̃∗ (t) . It follows from combining (3.20) and (3.21) that
∇η̃ ~ (t, q)
~ ~∗
η̃∗ (t, q)
∼
F̃ (φ̃∗ ) − S̃(φ̃∗ ) · K̄0 + K̄1 ~ + o(~)
∼ − G̃(φ̃∗ ) + o(~)
exp ~1 F̃ (φ̃∗ ) − S̃(φ̃∗ ) · (K0 + o(1))
− exp
1
~
as ~ → 0, which completes the proof.
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